Create bounding line around contour lines

I have a set of contour lines that I generated from processed LiDAR of a study area I am looking at in Central Washington. I want to convert the contours to polygons using the "Feature to Polygon" tool in Arc Map, but the edges of the contour data set are not closed, so when I try to create polygons from the contours, there are large gaps. Is there any way to create a bounding line around the outside of the contours to effectively seal all the gaps without doing it by hand? I have included to pictures to illustrate what I am talking about.